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 "cells": [
  {
   "cell_type": "markdown",
   "id": "8f283f78",
   "metadata": {},
   "source": [
    "### 11.1.1. 优化的目标"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "3fa5117d",
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import mindspore\n",
    "from mpl_toolkits import mplot3d\n",
    "from d2l import mindspore as d2l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "b6f77650",
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return x * d2l.cos(np.pi * x)\n",
    "\n",
    "def g(x):\n",
    "    return f(x) + 0.2 * d2l.cos(5 * np.pi * x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "3ad4c311",
   "metadata": {},
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      "text/plain": [
       "<Figure size 252x180 with 1 Axes>"
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     "metadata": {
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     "output_type": "display_data"
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   ],
   "source": [
    "def annotate(text, xy, xytext):  #@save\n",
    "    d2l.plt.gca().annotate(text, xy=xy, xytext=xytext,\n",
    "                           arrowprops=dict(arrowstyle='->'))\n",
    "\n",
    "x = d2l.arange(0.5, 1.5, 0.01)\n",
    "d2l.set_figsize((4.5, 2.5))\n",
    "d2l.plot(x, [f(x), g(x)], 'x', 'risk')\n",
    "annotate('min of\\nempirical risk', (1.0, -1.2), (0.5, -1.1))\n",
    "annotate('min of risk', (1.1, -1.05), (0.95, -0.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "12248265",
   "metadata": {},
   "source": [
    "### 11.1.2. 深度学习中的优化挑战"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd18bf33",
   "metadata": {},
   "source": [
    "#### 11.1.2.1. 局部最小值"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "ce0aa290",
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 252x180 with 1 Axes>"
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     "output_type": "display_data"
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   ],
   "source": [
    "x = d2l.arange(-1.0, 2.0, 0.01)\n",
    "d2l.plot(x, [f(x), ], 'x', 'f(x)')\n",
    "annotate('local minimum', (-0.3, -0.25), (-0.77, -1.0))\n",
    "annotate('global minimum', (1.1, -0.95), (0.6, 0.8))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de98d8bf",
   "metadata": {},
   "source": [
    "#### 11.1.2.2. 鞍点"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "d4559a12",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
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   "source": [
    "x = d2l.arange(-2.0, 2.0, 0.01)\n",
    "d2l.plot(x, [x**3], 'x', 'f(x)')\n",
    "annotate('saddle point', (0, -0.2), (-0.52, -5.0))"
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  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "62923097",
   "metadata": {},
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      "text/plain": [
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   ],
   "source": [
    "x, y = d2l.meshgrid(\n",
    "    d2l.linspace(-1.0, 1.0, 101), d2l.linspace(-1.0, 1.0, 101))\n",
    "z = x**2 - y**2\n",
    "\n",
    "ax = d2l.plt.figure().add_subplot(111, projection='3d')\n",
    "ax.plot_wireframe(x, y, z, **{'rstride': 10, 'cstride': 10})\n",
    "ax.plot([0], [0], [0], 'rx')\n",
    "ticks = [-1, 0, 1]\n",
    "d2l.plt.xticks(ticks)\n",
    "d2l.plt.yticks(ticks)\n",
    "ax.set_zticks(ticks)\n",
    "d2l.plt.xlabel('x')\n",
    "d2l.plt.ylabel('y');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6461a6af",
   "metadata": {},
   "source": [
    "#### 11.1.2.3. 梯度消失"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "76829585",
   "metadata": {},
   "outputs": [
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   ],
   "source": [
    "x = d2l.arange(-2.0, 5.0, 0.01)\n",
    "d2l.plot(x, [d2l.tanh(x)], 'x', 'f(x)')\n",
    "annotate('vanishing gradient', (4, 1), (2, 0.0))"
   ]
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   "id": "9c531836",
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